The Distribution of Group Structures
on Elliptic Curves over Finite Prime Fields

We determine the probability that a randomly chosen elliptic curve $E/{\F}_p$
over a randomly chosen prime field ${\F}_p$ has an ${\ell}$-primary part
$E({\F}_p) [\ell^{\infty}]$ isomorphic with a fixed abelian $\ell$-group
$H^{(\ell)}_{\alpha,\beta} = {\Z}/{\ell}^{\alpha} \times {\Z}/\ell^{\beta}$.
\smallskip
Probabilities for ``$|E(\F_p)|$ divisible by $n$'', ``$E(\F_p)$ cyclic''
and expectations for the number of elements of precise order $n$ in $E(\F_p)$
are derived, both for unbiased $E/\F_p$ and for $E/\F_p$ with $p \equiv
1~(\ell^r)$.